YouSaid · the spoken record

Joel David Hamkins

lines on the record: 286
first: 2025-12-31
most recent: 2025-12-31
sittings or episodes: 1
sources: podcast

Every line below is reproduced as it was said and linked to the record it came from. Nothing here is summarised or generated. Directory · Search · Corrections

“What are the techniques? So, one thing is that because of the incompleteness theorem, we know that this going And so we're already aware of the fact that there will always be these independence phenomenon for any theory that we write. And furthermore, some of those theories, we won't even be able to prove that they're consistent, like the consistency of the own theory. So that's called the consistency strength hierarchy. So it's a direct consequence of Girdle's second incompleteness theorem that for any theory we can write down then towering over it is this incredibly tall tower of consistency, strength, where the strength and theories aren't just adding another axiom, but they're adding another axiom even whose consistency was not provable in the previous layers of the hierarchy.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“But my attitude and the attitude of all the Sithirists was when you ask a question that turns out to be independent, then you asked exactly the right question because this is the one, you know, it's carving nature at its joints. You're adjudicating the nature of setheatic reality by finding these two realms, you find one of these dichotomies. There's the worlds where it's true and the worlds where it's false. And so when you ask that question, that's to be celebrated. It means you asked exactly the right, interesting, fascinating question. So it's not a kind of bleak thing that you can't prove it and you can't refute it. And that's such a disaster. Rather, it means that you found this This cleavage in reality, in mathematical reality, and it's good to know about those when they happen.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“But that's an interesting way to put it, I think, because it reminds me of this when I was a graduate student in Berkeley, there was another graduate student who was working with a non-logic professor in C-STAR algebras or something like this. So it's a part of analysis or functional analysis. And they were looking at a question and it turned out to be independent of CFC, right? And the attitude of this other professor was that, oh, I guess I asked the wrong question.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Right, exactly. So to be independent means you can't prove it, and also you can't prove that it's false. And you were saying it.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“You have a theory and it doesn't answer any of the questions that you're interested in. Okay, so what does that mean? If you're following what I call the universe view or the monist view You might naturally say, well, look, ZFC is a weak theory, and there's the true set theoretic reality out there, and we need a better theory because the current theory isn't answering the questions. Everything's independent. And so that seems like a quite reasonable thing to take. If you think that every set theoretic question has a definite answer and there's a unique set theoretic truth or a unique fact of the matter, right? This is the universe view”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Yeah, sure. Let's get into it. So the lesson of Cohen's result and girdle's result and so on, he's producing these alternative set theoretic universes We've observed that the continuum hypothesis is independent and the axiom of choice is independent of the other axioms. But it's not just those two. We have thousands of independence results. Practically every non-trivial statement of infinite combinatorics is independent of CFC. I mean, this is the fact. It's not universally true. There are some extremely difficult prominent results where people proved things in CFC. For the most part, if you ask a non-trivial question about infinite cardinalities, then it's very likely to be independent of CFC. And we have these thousands of arguments, these forcing arguments that are used to establish that. And so how should we take that? I mean, on the one hand,”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Giving us this extremely powerful tool for building alternative mathematical realities is how I think about it. He's explained to us how to take any set theoretic world and build another different one in which the condition hypothesis is false, the forcing extension.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“To build this alternative set theoretic reality, the constructual universe. And then he proves that the aximo choice is true there and also the continuum hypothesis is true there. And it's just amazing. Really beautiful argument. Okay, so then for the other part of the independence, that's only half of it because Girdle shows basically that you can't refute the continuum hypothesis. But that's not the same thing as proving that it's true. He showed that If set theories consists in without the continuum hypothesis, then it's consistent with the continuum hypothesis. So that's not the same thing as proving that it's true. And then it didn't come until 1963 when Paul Cohen invented the method of forcing and proved that if this a model of set theory, then this a model of set theory in which the continuum This is false So, Cohen also is”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“So he solved this. This is the same result where he answers the safety question of the axiom of choice, but also for the continuing hypothesis. They're true in the same center at a universe. We get, so if ZF without the axial choice is consistent, then so is ZFC plus the continuum hypothesis is the result. 1938. It's really such a beautiful argument. It's just incredible, I think, because he's building an alternative mathematical reality. That's the structure of the proof, is that, okay, if there's any mathematical reality, if there's any set theoretic world, then we're going to build another one, a separate one, a different one, maybe different. Maybe it's the same as the original one. It could be if we started already in the one that he built, then it would be the same. But there's no reason to assume it was the same. So he has this kind of model construction method.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“And then it's totally open. Hilbert asks about it at the turn of the 20th century. Nobody has any clue. There's no answer coming. Until 1938, this is four decades later, right? So a long time. And Kurt Girdle proved that. Half of it, what he proved is that If the axioms of set theory are consistent, then there is a set theoretic world where both the axiom of choice and the continuum hypothesis are true. So, what he's doing is showing this is called the constructible universe, girdles L.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“So he's sort of presuming that there is an algorithm, but he wants to know what it is, what is the algorithm. But the problem was solved by proving that there is no algorithm. It's an undecidable problem like the halting problem. There is no computable procedure that will correctly decide whether a given polynomial equation has a solution in the integers. So that's quite a remarkable development, I think. So there's a few other logic related questions on the list.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Core fundamental question to ask. So it seems quite natural that he would put it on the list. There were other logic related questions, though, like Hilbert's tenth problem is also related to logic. This is the question about Diophantine equations. And he asked to provide an algorithm to decide whether a given Deafantine equation has solution in the integers. So a Diophantine equation is just, I mean, it's a maybe a fancy way of talking about something that's easy to understand, a polynomial equation, except it's not just one variable, many variables. So you have polynomials in several variables over the integers, and you want to know, can you solve it? So the problem is, as stated by Hilbert, Provide an algorithm for answering the question whether a given polynomial equation has a solution in the integers.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“So that must have been part of Hilbert's thinking about why it's so important to have a uniform foundation and set theory was playing that role at the time. Now, of course, we have other possible foundations coming from category theory or type theory. And this univalent foundations now. So this sort of competing foundations now, there's no need to just use one foundation, one set theoretic foundation, although set theory continues to, in my view, have an extremely successful meta-mathematical analysis as a foundation, I think, is much more successful in set theory for any of those other foundations, but it's much less amenable, though, to things like computer proof and so on, which is part of the motivation to find these alternative foundations. So, yeah, okay, so just talk about Hilbert, though. I think he was motivated by the need for a unifying foundation of mathematics and said theory was playing that role and the continuum hypothesis is such a”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“The proof methods for that theorem come from other parts of mathematics, you know, this topological proofs and so on. And so what is that, how does that work? I mean, if you have totally different axiom systems, but you're using results from one subject in another subject, it's somehow incoherent unless this one underlying subject. So the unity of mathematics was provided by the existence of a mathematical foundation like set theory, and at the time it was set theory. And so it's critically important to be able to have a single theory in which one views all of mathematics as taking place to resolve that kind of transfer and borrowing phenomenon that was definitely happening.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“With their own axioms, separate axioms, right? But sometimes it happens like when you're proving, say, the fundamental theorem of algebra, you know, that the complex numbers are an algebraically closed field that you can solve any polynomial equation in.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“And we already discussed how Hilbert's views on the nature of set theory and the fundamental character that where he said no one will cast us from the paradise that Cantor has created for us. So I think Hilbert was convinced by Cantor on the importance and the fundamental nature of the continuum hypothesis for the foundations of mathematics, which was a critically important development for the unity of mathematics. I mean, before set theory emerged as a foundation of mathematics, there were, you know, this different subjects in mathematics, this algebra and this analysis, real analysis and topology and geometry. And so there's all these disparate subjects.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“I find it a little hard to believe that Hilbert would have conceived of his list in the same way that we now take his list. I mean, having observed the century unfold, we know that that list of twenty three problems did in fact guide whole research programs, and it was extremely important and influential. But at the time, Hilbert would have no reason to think that that would be true. And he was just giving a lecture and had a list of problems that he thought were very important. And so I would find it more reasonable to think that he was just making a list of problems that he thought were extremely interesting and important and fundamental in a way without the kind of heavy burden of guiding this 20th century research, although it turns out that, in fact, that's exactly what they did.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“So, right. So Hilbert had introduced at his famous address at the turn of the century these list of problems that he thought could guide or were important to consider in the coming century of mathematics. I mean, that's how people talk about it now, although I'm not sure at all. Of course, I can't really speak for Hilbert at all, but if you were a very prominent mathematician,”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“And therefore, you don't need the atoms if you're a structuralist because you only care about the structures up to isomorphism anyways. And the theory is simply more elegant and clearer without the atoms. They're just not needed. And so that's why today when we talk about set theory generally, we talk about the atom free version and ZFC has no ear elements. Okay, so we formulate the CFC axioms of set theory. These are expressing the main principal ideas that we have about the nature of sets and set existence. Cantor had asked about the Canadian hypothesis. In the late 19th century. And it remained open, totally open. Until nineteen thirty eight.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“I argue that it's really the philosophy of structuralism that leads them to omit the earle elements because it turns out that if you adopt ZFC axioms with earle elements ZFCU it's called or ZFA. Than any structure that exists, any mathematical structure that exists in that set theoretic universe with the atoms is isomorphic to a structure that doesn't use the atoms at all.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“So the ZFC axioms were, the axioms that were put forth first by Zermello in nineteen oh eight in regard to his proof of the well order using the axiom of choice. That wasn't Feli ZFC at that time, it was just Zermello theory because he sort of, there was a kind of missing axiom, the replacement axiom, and the foundation axiom were added. Axiomization, which became sort of standard. Actually, there's another aspect which is Zermelo's original theory allowed for the existence of Er elements or these atoms, mathematical objects that are not sets, but out of which we build the set theoretic universe, whereas set theorists today generally don't use Er elements at all”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Even with what we know now, it hasn't fully succeeded, and it can't because the hierarchy of complexity doesn't include all sets of real numbers. Some of them are sort of transcending this arche completely, in a way. And so the program can't ever fully be successful, especially in light of the independence results.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Realms even at the level of projective hierarchy, which are sets that you can define by using quantifiers over the real numbers themselves. So you get this hierarchy on top of the Borel hierarchy, the hierarchy of projectively definable sets. And it turns out that if you have enough large cardinals, then the projective sets also are Always either countable or equanumerous with the whole realign. And then one can try to go beyond this and so on. So I view all of those results which came, you know, in the past 50 years, the later ones, as fulfilling this candor idea that goes back, you know, one hundred and twenty years to his idea that we would prove the continuer, well, this is by establishing more and more instances for greater and greater complexity of sets. But of course,”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“And candor proved with a clothes that no, it's impossible. Every close set is either countable or echonerous with the whole real line. What the CANR program for solving the continuum of all this was. Sort of working up. So you did it for OpenSets and for closed sets, and you sort of work up maybe he wants to go into what are called the BREL sets, which are sort of combinations of open and closed sets. And there's a vast hierarchy of Borel complexity. And it turns out that the continuum hypothesis has been proved also for the Borel sets in this hierarchy. But then one wants to go beyond. What about more complicated sets? So there's this hierarchy of complexity for sets of real numbers and Cantor's idea was to sort of work your way up the hierarchy by proving that the continuing hypothesis was more and more true for those more and more complicated sets based on our understanding of the earlier cases. And that has been carried out to a remarkable degree. It turns out that one needs, one begins to need large cardinal assumptions, though, in order to get to the higher”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Sequences that are converging to a point that would also be a closed set, or a convergent sequence is of convergent sequences and so on. That would be a closed set also.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Oh, yeah, sure. So a set of reels is open if every point that it contains is surrounded by a little interval of points, the whole tiny little interval. But that tiny little interval is already just by itself equinormous through the whole line. So that's why the question is sort of easy for OpenSet. A closed set is a complement of an open set. And there's a lot of closed sets that are really complicated of varying sizes. So of course any closed interval is a closed set, but it's not only those. There's also things like the cantor set, which you get by omitting middle thirds. Maybe some people have seen this construction. Or you can imagine sort of randomly taking a lot of little tiny open intervals all over the line and so on. So that altogether would be an open set and the complement of it would be a closed set. So you can imagine just kind of tossing down these open intervals and what's left over is the closed set. Those sets can be quite complicated. And they can have isolated points, for example, if the two open intervals were just kissing and leaving only the one point between them. But also you could have”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“So he had a program for proving it, which has been affirmed in a certain respect. Of course, the Canadian hypothesis holds for open sets. That's easy to see. If you have an open interval, then this is fully equanumerous with the whole real line. Any interval is equinumerous with the whole line, because all you would need is a function, you know, like the arc tangent function or something that maps the whole real line into an interval. And that's a one-to-one function. So we know the open sets have the property that their non-trivial open sets are all fully equanimous with the whole real line. So never strictly in between. But remarkably, Kenner proved it also for the closed sets. And that is using what's called the Kannerbendixon theorem. So it's quite a remarkable result. It's definitely not obvious. And in this theorem actually was the origin of the ordinals. Cantor had”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“But it turned out that for all the sets that anyone ever could define or pick out or observe, for all the sets of real numbers, it was always the case either that they were countable in which case their equanumerous with the natural numbers are else finite, or they were fully equinumerous with the whole real line. And so they were never strictly in between. I mean, you're in this situation and you have hundreds, thousands of sets that are candidates to be in between, but in every single case, you can prove it's on one side or the other and not strictly in between. And so in every situation where you're able to figure out whether it's in between or not, it's always never strictly in between.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“I mean, what could be a more natural question to ask immediately after that? And so Cantor did ask it, and he spent his whole life thinking about this question. So the continuum hypothesis is the assertion that there is no infinity in between the natural numbers and the real numbers. And of course, Cantor knew many sets of real numbers. Everything in between, I mean, everything that's in that interval would be equinumerous with some set of real numbers. But we know lots of sets of real numbers. I mean, there's all these various closed sets, Canner sets, and so on, this Vitali said. We have all kinds of sets of real numbers. And so you might think, well, if the continued hypothesis is false, then we probably seen the set already. We just have to prove that it's strictly in between.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“So, the continuum of all this is the question that arises so naturally whenever you prove that there's more than one size of infinity. So candor proved that the infinity of the real numbers is strictly larger than the infinity of the natural numbers. But immediately when you prove that, Wants to know well, is there any?”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“By learning enough about that other subject matter, this is what was so rewarding for me because basically I had to learn enough, my expertise, my main expertise was logic, but someone would ask a question that was about, say, the axiom of choice in this other subject matter or the continuum hypothesis or something like that in the other subject matter. And I would have to learn enough about that other subject in the context of the question in order to answer. And I was often able to do that. And so I was quite happy to do that. And also I learned a lot by doing that because I had to learn about these other problem areas. And so it really allowed me to grow enormously as a mathematician.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“So, I think when I first joined Math Overflow, I was basically one of the only, one of the few people in logic who was answering. I mean, there were other people who know some logic, particularly from category theory and other parts of mathematics that aren't in the most traditional parts of logic, but they were answering some of the logic questions. So I really found myself able to make a contribution in those very early days by engaging with the logic related questions, but there weren't many logic people asking questions either. But what I found was that there was an enormous amount of interest in topics that were logic adjacent. So a question would arise, you know, in group theory, but it had a logic aspect or an analysis or whatever, and there would be some logic angle on it. And what I found was that I was often able to figure out an answer.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“So I'm interested in any question that I find interesting. And it's not all questions. Sometimes certain kinds of questions just don't appeal to me that much.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Well, math overflow has really been one of the great pleasures of my life. I've really enjoyed it. I mean, and I've learned so much from interacting on math overflow. I've been on there since 2009, which was shortly after it started. I mean, it wasn't exactly at the start, but a little bit later. And I think it gives you the stats for how many characters I typed, and I don't know how many million it is, but this enormous amount of time that I've spent thinking about those questions. And it has really just been amazing to me.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Would probably be completely unrecognizable to me. I maybe wouldn't even begin to understand what they're talking about even without sort of witnessing the intervening developments. So if you bring someone from ancient times to today, they maybe wouldn't even understand what we're talking about with some of the questions. But I feel that... If Archimedes came and we were able to communicate, I think I would be able to tell him about some of the things that are going on in mathematics now and maybe. Or anyone from that time, I mean. So I think it is possible to have this kind of progress, even when the subject kind of shifts away from the earlier concerns. As a result of the progress, basically.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“The field sort of moves on to more difficult, interesting questions. Whereas in philosophy this a little bit true that there's progress, but meanwhile, it's also true that there are these eternal questions that have been with us for thousands of years. And in fact, so much so that you can find a lot of philosophers arguing that the important contribution of philosophy is in asking the questions rather than answering them. Because it's hopeless to answer them. I mean, the nature of these deposit questions is so difficult, less of a sense of progress is what I'm trying to say. I don't see any reason to think that the progress in mathematics in the growth in our mathematical understanding and knowledge won't simply continue. And so a thousand years from now, maybe the mathematics that they will be doing at that time”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“It's interesting to me because I have my feet into worlds of mathematics and philosophy and to compare the differences between these subjects. And one of the big, there's many cultural differences, but one of the big cultural differences is towards the idea of progress in the subject. Because mathematics has huge progress. We simply understand the mathematical ideas much, much better continually improving our understanding and there's growth in knowledge. We understand the nature of infinity now better than they did a hundred years ago. I mean, definitely better. And they understood it better a hundred years ago than they did, you know, for the previous thousands of years and so on. So in almost every part of mathematics, there's improved understanding of the core issues so much so that the questions at hand become totally different.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Maybe I could hope that someone will give the convincing account, but it seems to be a profound mystery to me. I can't even imagine what it would be like to give an account of physical existence.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“But I don't think we actually have an understanding at all, I mean very, very little of the nature of physical existence. I think it's a profound mystery. Whereas I think that we do have something a little better of an understanding of the nature of mathematical existence and abstract existence. So that's how I would describe the point.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“The mathematical platonic realm is I'm not sure I would say it's more real, but I'm saying we understand the reality of it in a much deeper and more A more convincing way. I don't think we understand the nature of physical reality very well at all. And I think most people aren't even scratching the surface of the question as I intend to be asking it. So, you know, obviously we understand physical reality. I mean, I knock on the table and so on and we know all about what it's like to, you know, have a birthday party or to drink a martini or whatever. And so we have a deep understanding of existing in the physical world. But maybe understanding is the wrong word. We have an experience of living in the world and riding bicycles and all those things.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Do abstract objects exist in a place and at a time? That's very debatable, I think. And what does place and time mean? Yeah.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Oh, yeah, totally. Yeah. This is the realist position in mathematics that abstract objects have a real existence. And, okay, what's meant by that is that there's some sense of existence in which those objects can be regarded as real.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“I'm not quite sure. I mean I live entirely in the Platonic realm, and I don't really understand the physical universe at all, so I don't have strong views.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Is that Julius Caesar is not a number The structuralists disagree with that position. The structuralist attitude is, look, you give me a number system. If Julius Caesar isn't a number, then I can just let's take the number seventeen out of that system and plug in Julius Caesar for that role. And now I've got a new number system, and now Julius Caesar happens to be the number 17. And that's totally fine. Yeah, so the point of structuralism is that the question of whether Julius are a number or not is irrelevant to mathematics. It is irrelevant because it is not about structure. It's about this essence of the mathematical objects. So that's the structuralist criticism of Frege's point.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“That well, there was something that dissatisfied him about that situation, which is that the Canerhume principle Does not seem to give you a criteria for which things are numbers. It only tells you a kind of identity criteria for when are two numbers equal to each other. Well, two numbers are equal just in case the sets of those sizes are equanumerous. So that's the criteria for number identity. But it's not a criteria for what is a number. And so this problem has become known as the Julius Caesar problem because Frege said, we don't seem to have any way of telling from the Hume principle whether Julius Caesar is a number or not. So he's asking about the essence of number and whether, of course, one has the sense that he picked maybe what he was trying to present as a ridiculous example because maybe you have the idea that, well, obviously Julius Caesar is not a number and there's a lot of philosophical writing that seems to take that line also that obviously the answer.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Yeah, I think so, because I guess part of the point of structuralism is that it doesn't make sense to consider mathematical objects or individuals in isolation. What's interesting and important about mathematical objects is how they interact with each other and how they behave in a system. And so maybe one wants to think about the structural role that the objects play in a larger system, a larger structure. There's a famous question that Frege had asked, actually, when he was looking into the nature of numbers, because in his logic program, right, he was trying to reduce all mathematics to logic. And in that process, he was referring to the Kanner-Hume principle that, you know, whenever two sets are equinumerous, then they have the same number of elements. I mean, if and only if. And he founded his theory of number on this principle, but he recognized”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Of the properties of this systems number four with regard to any question that's important about the number four. But those questions won't be about essence. So in a sense, structuralism is a kind of anti-essentialism in mathematics.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“You know, in a mathematical structure is irrelevant with regard to any mathematical property of that structure. And so So, to ask a question, like, what is the number four really? Is an anti-structuralist thing? Because if you have a structure, say the natural numbers, you know, with all the numbers in at 0, 1, 2, 3, 4, and so on, then I could replace the number four with something else like this bottle of water could play the role of the number four in that structure. And it would be isomorphic. And it wouldn't matter at all for any mathematical purpose to use this alternative mathematical system. That's to say that we don't care what the number four is really. That is irrelevant. The only thing that matters is what are the properties of the number four in a given mathematical system, you know, and recognizing that there are other isomorphic copies of that system and the properties of that other system's number four are going to be identical to the property.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Structuralism is a philosophical position in mathematics or the philosophy of mathematics by which one emphasizes that what's important about mathematical objects is not what they're made out of or what their substance or essence is, but rather how they function in a mathematical structure. And so what I call the structuralist attitude in mathematics is that we should only care about our mathematical structures up to isomorphism. If I have a mathematical structure of a certain kind and I make an exact copy of it using different individuals as to form the elements of that structure, then the isomorphic copy is just as good mathematically and there's no important mathematical difference that would ever arise from working with this isomorphic copy instead of the original structure. And so therefore, that's another way of saying that the substance of individual”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Well, there's different ways to understand the nature of four. I mean, actually, this gets into the question of structuralism, which is maybe a good place to talk about it.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source